The Eighth International Conference on Science and Mathematics Education in Developing Countries (Part 4) Dr Khin Maung Win

The  Eighth International Conference on Science and Mathematics Education in     Developing  Countries

Part (4)

                                      Dr Khin Maung Win 

 

(Arts Assembly)

The next paper was read by Dr. Aye Pyone of Mathematics Department , Yan University.It was called Chinese Remainder Theorem  and Its Applications.Remainder Theorem ) without the adjective Chinese ) is part of the  10 th standard course. The Chinese Remainder Theoremis part of the Number Theory.I remember what my teacher said about this theory.Hs said – those who want to make a name in mathematics with just a knowledge of the four basic operations,addition,subtraction, multiplication,division   should study number theory, where there are still many unsolved problems whose solutions have evaded the experts.Saying that,he recounted to us the way Euler answered Fermats question which has beaten the experts for more than  a hundred years. 

 

(At Arts Assembly)

   In order to understand the statement of the Chinese Remainder Theorem  on must know the definition of relatively prime  and the meaning of the equation   x = y mod m  . Two numbers are relatively prime means that the only number which divides both of them is one.For example  6 and 7 are relatively prime    but  6 and 8  are not.

 The equation     x = y mod m    means that    x – y   or  y-x   is divisible  by  m . For example     7  =  13  mod 2    ., since   13 – 7   =  6     is divisible  by   2 .

The statement of the theorem is as follows ; If

 b₁,b₂,b₃, ……..are positive integers  and    m₁,m₂,m₃,………   are pairwise relatively prime,then there exists  unique  x   which satisfies the equations     x = b_i mod mi  ,   for   i= 1,2,3,…..n.

 

(I am with  a professor)

This statement is called Chinese Remainder Theorem.In the conclusion it is stated that   there exists a unique   x   such that    x-b₁,x-b₂, x-b₃ ,  ……    is divisible by  m1,m2,m3,  ……….It assures us of the existence of  x  ,as most existence  theorems do, but it does not tell us how it find it. The following example illustrates that point.

 

(With Myanmar Professors at front.)

Two express trains depart from Yangon to Mandalay  at  40 pph   and  43  mph  respectively.They remain  at  6   and  22 miles  to reach Mandalay  respectively  after travelling  a few hours, . Find the distance between Yangon and  Mandalay.

             Yangon  x---------------------------------------x Mandalay

                                    40 mph        ,  43 mph  

    Let   the distance between   Yangon and Mandalay  be   x   miles.   

          x – 6     is   divisible  by    40 .

           x – 22   is    divisible    by   43   

   In Chinese Reminder Theorem  we take        b₁ =  6   ,   b₂=  22 

                                                                           n₁ = 40   ,  n₂ =  43

    Then   40  and  43   are relatively prime,. Hence by the theorem, there exists a unique  x   such  that  x-6  is divisible  by  40   and   x – 22       is divisible by  43.

    Hence   for some  integers   a   and   b  ,   x-6 =40a  and    x-22 =  43b .

   Hence   40a + 6  = 43b + 22 =  x            …………#

 We try to find  the values of the  expressions  40a+6   and  43b+ 22   by trial and error to find   x    whose existence and uniqueness are assured by the theorem.

                         40a + 6                    43b + 22

                   a=1,      46                     b= 1   ,   65

                    a=2    ,    86                   b=2  ,   108

                    ………………………………….

             a= 9    ,  366                        b = 8   ,    366

According the trial and error,    When  a = 9   and   b=8 ,   x= 366 miles.

 

(With my friends professors.)

On the third day afternoon,there was a panel discussion and closing ceremony.The questions  to be discussed at the panel discussion were asked  for to be given in advance.

   When the time came,there were  eight professors seated on the stage.They took turns to talk about which I did not understand a word. I think it was a waste of time.After a waste of one hour,the questions from the aucience were shown on the screen.I do not remember the questions of the others. I can only relate to the readers the question I posed and how it was answered.My question was as follows

;        In the teaching of problem solving skills, how much emphasis should be given to the speed with which the solution is  arrived at ? Choose one of the answers.   1.  A little     2.   Much      3. Not at all .

    My own opinion is  number  3 , not at all .

 

(I am at a lunch.)

In answering the question  a talk on the objectives of teaching of problem solving without giving the direct answer to the questAt the closing ceremony,the president announced that the next conference would be held next year at Mandalay.

     That is the end of  my experience at the conference.  

Dr' Khin Maung Win

Retired Professor of Mathematics 

Rangoon University

Burma